Uniform distributions and random variate generation over generalized [formula omitted] balls and spheres

Abstract

This paper studies a class of distributions supported on generalized balls and spheres that are extensions of lp balls and spheres, defined based on power-sum polynomials—which can be non-symmetric. The joint, marginal, and conditional distributions corresponding to these distributions are presented. Then, three stochastic representations are developed based on independent uniform distributions, multivariate Dirichlet distributions, and independent generalized normal (Gaussian) distributions. The sampling schemes developed based on these representations are numerically compared. Moreover, it is shown that the distributions of any two associated generalized lp ball and sphere can be represented based on each other. It is observed that the proposed distributions over generalized lp balls are uniform with respect to the Lebesgue measure. However, except in the case of lp spheres, the distribution over a generalized lp sphere is not uniform with respect to the corresponding cone measure (a disintegration of the Lebesgue measure on the associated generalized ball). Using our results, a procedure is provided for uniform sampling over generalized-sphere manifolds with respect to the Hausdorff measure—which is known as the natural surface-area measure. The proposed distributions can be used to define new classes of distributions and copulas that can be sampled efficiently. Moreover, our results pave the way to define flexible and manageable uncertainty sets in robust optimization, or in analysis and control of uncertain systems.